Skip to main content Accessibility help


  • Javier Hidalgo (a1) and Myung Hwan Seo (a1)


We consider an omnibus test for the correct specification of the dynamics of a sequence $\left\{ {x\left( t \right)} \right\}_{t \in Z^d } $ in a lattice. As it happens with causal models and d = 1, its asymptotic distribution is not pivotal and depends on the estimator of the unknown parameters of the model under the null hypothesis. One first main goal of the paper is to provide a transformation to obtain an asymptotic distribution that is free of nuisance parameters. Secondly, we propose a bootstrap analog of the transformation and show its validity. Thirdly, we discuss the results when $\left\{ {x\left( t \right)} \right\}_{t \in Z^d } $ are the errors of a parametric regression model. As a by product, we also discuss the asymptotic normality of the least squares estimator of the parameters of the regression model under very mild conditions. Finally, we present a small Monte Carlo experiment to shed some light on the finite sample behavior of our test.


Corresponding author

*Address correspondence to Myung Hwan Seo, London School of Economics, Houghton St., London WC2A 2AE, UK; e-mail:; or to: Seoul National University, Kwan-Ak Gu, Seoul, Korea.


Hide All
Baltagi, B., Kelejian, H.H., & Prucha, I.R. (eds.) (2007) Analysis of spatially dependent data. Annals issue. Journal of Econometrics, Annals Issue, 140.
Batchelor, L.I.A. & Reed, H.S. (1924) Relation of the variability of yields of fruit trees to the accuracy of field trials. Journal of Agricultural Research 12, 245283.
Besag, J. (1974) Spatial interaction and the statistical analysis of lattice systems (with discussion). Journal of the Royal Statistical Society, Series B 36, 192236.
Bickel, P.J. & Wichura, M.J. (1971) Convergence criteria for multiparameter stochastic processes and its applications. Annals of Mathematical Statistics 42, 16561670.
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.
Bloomfield, P. (1973) An exponential model for the spectrum of a scalar time series. Biometrika 61, 217226.
Bolthausen, E. (1982) On the central limit theorem for stationary mixing random fields. The Annals of Probability 10, 10471050.
Brillinger, D.R. (1981) Time Series, Data Analysis and Theory. Holden-Day.
Brown, R.L., Durbin, J., & Evans, J.M. (1975) Components of Cramér-von Mises statistics II. Journal of the Royal Statistical Society, Series B 37, 216237.
Chen, H. & Romano, J.P. (1999) Bootstrap-assisted goodness-of-fit tests in the frequency domain. Journal of Time Series Analysis 20, 619654.
Cressie, N.A.C. (1993) Statistics for Spatial Data. Wiley.
Cressie, N. & Huang, H.C. (1999) Classes of nonseparable, spatio-temporal stationary covariance functions. Journal of the American Statistical Association 94, 13301340.
Crujeiras, R.M., Fernandez-Casal, R., & Gonzalez-manteiga, W. (2008) An L 2-test for comparing spatial spectral densities. Statistics & Probability Letters 78, 25432551.
Dahlhaus, R. & Künsch, H. (1986) Edge effect and efficient parameter estimation for stationary random fields. Biometrika 74, 877882.
Davis, R.A., Alexandre Trindade, A., & Breidt, F.J. (2001) Least absolute deviation estimation for all-pass time series models. Annals of Statistics 29, 919946.
Delgado, M.A., Hidalgo, J., & Velasco, C. (2005) Distribution free goodness-of-fit tests for linear processes. Annals of Statistics 33, 25682609.
Delgado, M.A., Hidalgo, J., & Velasco, C. (2011) Bootstrap assisted specification tests for the Farima model. Econometric Theory 27, 10831116.
Eubank, R.L. & LaRicca, V. (1992) Asymptotic comparison of Cramér-von Mises and nonparametric techniques for testing goodness-of-fit. Annals of Statistics 20, 20712086.
Fernandez-Casal, R., Gonzalez-Manteiga, W., & Febrero-Bande, M. (2003) Flexible spatio-temporal stationary variogram models. Statistics and Computing 13, 127136.
Genton, M.G. & Koul, H.L. (2008) Minimum distance inference in unilateral autoregressive lattice processes. Statistica Sinica 18, 617631.
Giacomini, R., Politis, D.N., & White, H. (2013) A warp-speed method for conducting Monte Carlo experiments involving bootstrap estimators. Econometric Theory 29, 567609.
Grenander, U. (1950) Stochastic processes and statistical inference. Arkiv för Matematik 1, 195277.
Guyon, X. (1982) Parameter estimation for a stationary process on a 𝑑-dimension lattice. Biometrika 69, 95105.
Hainz, G. & Dahlhaus, R. (2000) Spectral domain bootstrap tests for stationary time series. Preprint.
Hannan, E.J. (1970) Multiple Time Series. Wiley.
Hannan, E.J. (1973) The asymptotic theory of linear time series models. Journal of Applied Probability 10, 130145.
Hidalgo, J. (2009) Goodness of fit for lattice processes. Journal of Econometrics 151, 113128.
Hidalgo, J. & Kreiss, J.P. (2006) Bootstrap specification tests for linear covariance stationary processes. Journal of Econometrics 133, 807839.
Hong, Y. (1996) Consistent testing for serial-correlation of unknown form. Econometrica 64, 837864.
Ibragimov, I.A. & Rozanov, Y.A. (1978) Gaussian Random Processes. Springer-Verlag.
Jenish, N. & Prucha, I.R. (2009) Central limit theorems and uniform laws of large numbers for arrays of random fields. Journal of Econometrics 150, 8698.
Khmaladze, E.V. (1981) Martingale approach in the theory of goodness-of-fit tests. Theory of Probability and Applications 26, 240257.
Lanne, M. & Saikonnen, P. (2011) Noncausal autoregressive for economic time series. Journal of Time Series Econometrics 3, 130.
Lanne, M. & Saikonnen, P. (2013) Noncausal vector autoregression. Econometric Theory 29, 447481.
Mardia, K.V. & Marshall, R.J. (1984) Maximum likelihood estimation of models for residual covariance in spatial regression. Biometrika 71, 135146.
McKeague, I.W., Nikabadze, A.M., & Sun, Y. (1995) An omnibus test for independence of survival time from a covariate. Annals of Statistics 223, 450475.
Mercer, W.B. & Hall, A.D. (1911) The experimental error of field trials. Journal of Agricultural Science 4, 107132.
Mitchell, M.W., Genton, M.G., & Gumpertz, M.L. (2005) Testing for separability of space–time covariances. Environmetrics 16, 819831.
Neyman, J. (1937) Smooth test for goodness of fit. Arkiv for matematik 20, 150199.
Paparoditis, E. (2000) Spectral density based goodness-of-fit tests for time series models. Scandinavian Journal of Statistics 27, 143176.
Robinson, P.M. (1988) The stochastic difference between econometric statistics. Econometrica 56, 531548.
Robinson, P.M. (2012) Inference on power law spatial trends. Bernoulli 18, 644677.
Robinson, P.M. & Thawornkaiwong, S. (2012) Statistical inference on regressing with spatial dependence. Journal of Econometrics 167, 521542.
Robinson, P.M. & Vidal-Sanz, J. (2006) Modified Whittle estimation of multilateral models on a lattice. Journal of Multivariate Analysis 60, 234349.
Schoenfeld, D.A. (1977) Asymptotic properties of tests based on linear combinations of the Cramér-von Mises statistic. Annals of Statistics 5, 10171026.
Shao, J. & Tu, D. (1995) The Jackknife and Bootstrap. Springer-Verlag.
Solo, V. (1986) Modeling of two-dimensional random field by parametric Cepstrum. IEEE Transactions on Information Theory 32, 743750.
Stute, W. (1997) Nonparametric model checks for regression. Annals of Statistics 25, 613641.
Whittle, P. (1954) On stationary processes in the plane. Biometrika 41, 434449.
Yajima, Y. & Matsuda, Y. (2011) Asymptotic properties of the LSE of a spatial regression in both weakly and strongly dependent stationary random fields. Preprint. University of Tokyo.
Zhu, J., Huang, H.C., & Reyes, P.E. (2010) On selection of spatial linear models for lattice data. Journal of the Royal Statistical Society, Series B 72, 389402.


  • Javier Hidalgo (a1) and Myung Hwan Seo (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.